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Question 1 Let 60 -67 37 A = 45 -51 27 26 -29 13 -4 -38 42 -26, and let V = RS. Define a linear operator T : V - V, T(x) = Ax. (a) Explain why if 1 is an eigenvalue of A with corresponding eigenvector v, then 1, the complex conjugate of 1, is an eigenvalue of A with corresponding eigenvector v. (3 marks) (b) Show that A has an eigenvalue 2i with corresponding eigenvector -46- 1 -36 + 3i -19+41 29 Prove also that A also has an eigenvalue -3 and find all the corresponding eigenvectors. (6 marks) (c) Determine T-invariant subspaces Wj and W2 of V such that V = W, @ W2. Give a basis of each subspace. (5 marks)(d) Let 10 -3 0 -3 0 0 B = -3 0 -4 -1 -10 0 Determine whether A and B are similar over the complex field C (6 marks)